I . . .have a child support case in process with my ex. However, over the last month I've decided that I want to move to a different state, closer to baby's father so they can possibly have a relationship.
Just to be clear, are your "ex" and the "baby's father" the same person?
I have a feeling if I told them I was moving that they would threaten a way to stop me (taking my little one, etc.) Is this something that would be a possibility?
Is it a possibility that they might threaten you? Of course it is (and, needless to say, you know them and we don't, so you obviously know better than anyone here what they might do). Likewise, it's possible that they might do any number of things beyond merely threatening you. Again, you know them and we don't.
I suspect what you really intended to ask is whether your parents have any legal right to prevent you from moving, and the answer to that is no (unless there are relevant facts you didn't include in your post).